ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

doi:10.1016/S0252-9602(13)60053-3

Abstract

Abstract:

A famous theorem of Szemer’edi asserts that any subset of integers with posi-tive upper density contains arbitrarily arithmetic progressions. Let F_q be a finite field with q elements and F_q((X⁻¹)) be the power field of formal series with coefficients lying in F_q. In this paper, we concern with the analogous Szemer´edi problem for continued fractions of Laurent series: we will show that the set of points x ∈ F_q((X−1)) of whose sequence
of degrees of partial quotients is strictly increasing and contain arbitrarily long arithmetic progressions is of Hausdorff dimension 1/2.

Key words: Szemer´edi theorem, continued fractions, Laurent series, Hausdorff dimension

CLC Number:

11K55

HU Dong-Gang, HU Xue-Hai. ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES[J].Acta mathematica scientia,Series B, 2013, 33(4): 943-949.

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